The three nations have been attempting to find a political solution to the Syrian conflict at talks that started last year in Astana, Kazakhstan, in competition with the US and UN-backed Geneva initiative.
The latest talks in Moscow between two of Assad's key supporters, Moscow and Tehran, and militant-backer Ankara come as the alleged chemical attack in the Syrian town of Douma on April 7 has prompted sharply differing responses from Turkey and Russia.
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"I curse those who carried out this massacre," President Recep Tayyip Erdogan said, adding that he welcomed Western air strikes in retaliation as 'appropriate'.
Meanwhile Russia says the attack was staged to discredit its ally President Bashar-al Assad and this week brought a group of Syrians to the global chemical arms watchdog to back its claims.
French President Emmanuel Macron this month suggested the air strikes had driven a wedge between Ankara and Moscow as they have been building increasingly close ties.
This prompted an angry denial from Foreign Minister Mevlut Cavusoglu, who said the countries' relations "are not so weak that the French president can break them".
Alexander Shumilin, a Middle East expert at the Institute for US and Canadian Studies in Moscow, however, said Douma fallout had "caused a crack in the alliance of three countries".
He added: "If the trio falls apart entirely, the ensuing events could be really bad."
Turkey has a "completely different attitude" to resolving the conflict and Assad's fate, he said, while Iran is seeking to destabilise the region to hurt Israel and Russia wants to "stabilise and leave".
Alexey Malashenko, a specialist in the Syria conflict, was sceptical the meeting would achieve much beyond a show of unity, saying the trio have a "very shaky" alliance.
Their positions on the suspected chemical attack are irreconcilable, he said.
"Turkey has an absolutely honest position: it is against Bashar al-Assad and here there's no way they can reach an agreement."
Nevertheless, the limited nature of US-led strikes suggests the "peak of tensions has passed", he said.
The Turkish foreign ministry said Saturday's meeting "will focus on all aspects" of cooperation and "elaborate on the steps that could be taken from now on", while the Russian foreign ministry said talks would focus on the humanitarian situation.
Russian Foreign Minister Sergei Lavrov will also hold bilateral talks, including on the "escalating situation" around the Iran nuclear deal, spokeswoman Maria Zakharova said.
The most recent Syria talks attended by the foreign ministers in Astana were held on March 16. The next are set for May 14.
EU diplomatic chief Federica Mogherini on Tuesday urged the three powers to do more, saying they "have not only a responsibility but also an interest in making the ceasefire work".
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Eight rounds of talks under United Nations auspices in Geneva have made little headway, with Assad's government taking little interest.
Both Tehran and Moscow have deployed military forces to Syria to back up Assad in his now seven-year-old war against anti-government militants.
Ankara has called for Assad's removal throughout the war, but has worked increasingly closely with Moscow and Tehran in recent months in an attempt to find a solution to the conflict.
In January, Russia held a showpiece conference in the Black Sea resort of Sochi with largely pro-regime delegates, which saw boycotts by militants and made little progress.
Russian President Vladimir Putin hosted a summit on Syria with the Iranian and Turkish presidents in November. They then met again in Ankara this month.
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